konstantin genin

Published and Forthcoming

This paper presents new logical relations connecting three topics pertaining to inductive inference: (I) synchronic norms of theory choice, like the preferences for simpler and more falsifiable theories, (II) diachronic norms of theory change familiar from belief revision and AGM theory, and (III) the justification of such norms by truth-conduciveness, or learning performance.

In topological learning theory, open sets are interpreted as hypotheses deductively verifiable by true propositional information that rules out relevant possibilities. However, in statistical data analysis, one routinely receives random samples logically compatible with every statistical hypothesis. We bridge the gap between propositional and statistical data by solving for the unique topology on probability measures in which the open sets are exactly the statistically verifiable hypotheses. Furthermore, we extend that result to a topological characterization of learnability in the limit from statistical data.

Glymour's early work on confirmation theory (1980) eloquently stressed the rhetorical plausibility of Ockham's razor in scientific arguments. His subsequent, seminal research on causal discovery (Spirtes et al. 2000) still concerns methods with a strong bias toward simpler causal models, and it also comes with a story about reliability---the methods are guaranteed to converge to true causal structure in the limit. However, there is a familiar gap between convergent reliability and scientific rhetoric: convergence in the long run is compatible with any conclusion in the short run. For that reason, Carnap (1945) suggested that the proper sense of reliability for scientific inference should lie somewhere between short-run reliability and mere convergence in the limit. One natural such concept is straightest possible convergence to the truth, where straightness is explicated in terms of minimizing reversals of opinion (drawing a conclusion and then replacing it with a logically incompatible one) and cycles of opinion (returning to an opinion previously rejected) prior to convergence. We close the gap between scientific rhetoric and scientific reliability by showing (1) that Ockham's razor is necessary for cycle-optimal convergence to the truth, and (2) that patiently waiting for information to resolve conflicts among simplest hypotheses is necessary for reversal-optimal convergence to the truth.

Ockham's razor says: "Choose the simplest theory compatible with the data." Without Ockham's razor, theoretical science cannot get very far, since there are always ever more complicated explanations compatible with current evidence. Scientific lore pretends that reality is simple---but gravitation works by a quadratic, rather than a linear, law; and what about the shocking failure of parity conservation in particle physics? Ockham speaks so strongly in its favor that demonstrating its falsity resulted in a Nobel Prize in physics (Lee and Yang 1957). So why trust Ockham?

We analyze log-data generated by an experiment with Fractions Tutor, an intelligent tutoring system. The experiment compares the educational effectiveness of instruction with single and multiple graphical representations. We cluster students by their learning strategy and find that the association between experimental condition and learning outcome is found among students implementing just one of the learning strategies. The behaviors that characterize this group illuminate the mechanism underlying the effectiveness of multiple representations and suggest strategies for tailoring instruction to individual students.

Accepted for Presentation

The distinction between deductive (infallible, monotonic) and inductive (falli-ble, non-monotonic) inference is fundamental in the philosophy of science. However, virtually all scientific inference is statistical, which falls on the inductive side of the traditional distinction. We propose that deduction should be nearly infallible and monotonic, up to an arbitrarily small, a priori bound on chance of error. A challenge to that revision is that deduction, so conceived, has a structure entirely distinct from ideal, infallible deduction, blocking useful analogies from the logical to the statistical domain. We respond by tracing the logical insights of traditional philosophy of science to the underlying information topology over possible worlds, which corresponds to deductive verifiability. Then we isolate the unique information topology over probabilistic worlds that corresponds to statistical verifiability. That topology provides a structural bridge between statistics and logical insights in the philosophy science.

Bayesian conditioning is widely considered to license inductive inferences to universal hypotheses. However, several authors [Kelly, 1996, Shear et al., 2017] have called attention to a sense in which those inferences are essentially deductive: if H has high credence after conditioning on E, then the material condition E ⊃ H has even higher prior probability. In this note, I show that a similar feature attends Jeffrey conditioning. Furthermore, I briefly address the extent of non-deductive undermining of prior beliefs.

We present and motivate a new explication of empirical simplicity that avoids many of the problems with earlier accounts. The proposal is grounded in information topology, the topological space generated by the set of all possible information states inquiry might encounter. Our proposal is closely related to Popper's, but we show that it improves upon his in at least two respects: maximal simplicity is equivalent to refutability and stronger hypotheses are not simpler. Finally, we explain how to extend the topological viewpoint to statistical inductive inference.